The organ that was nobody's neighbour

The family that lost a member is the family that survives, on every row where the question can be asked. That is nine rows of thirty. This essay is about the other twenty-one, and it is mostly an argument for leaving them alone.

An organ k places back from the tip lies on the tip’s own p-chain exactly when p divides k. On a stem counted at 5 and 8 spirals the answerable offsets are five and eight. Four, six and seven are not multiples of either, so the organ removed there is not a neighbour of the tip along any chain the counter names, and the reading has no purchase.

The proportion is not an accident of this lattice, and the arithmetic behind it is short enough to do here. Inside a front of depth q there are q offsets. The multiples of q among them number one, namely q itself. The multiples of the smaller number p number one as well — p, and then 2p, which is past the front, because the ratio of the two counted numbers is under two at every rung of either branch. So a lattice offers exactly two answerable offsets, whatever pair it carries. On a 5/8 stem they are five and eight, two of eight; the other six are silent. The twenty-one are what that arithmetic looks like summed over five lattices, and they would be twenty-one whatever the physics turned out to be. That is worth knowing before treating the gap as a finding about the rule rather than about the census’s shape.

What those rows do #

They wreck, and they keep a contact family like every other row. Offset four on a 5/8 stem keeps the five at almost every rise on its rung; offsets six and seven keep the eight. So there is structure — it is exactly the structure the offset rule describes — and the point is that the lost-member reading is not what produces it.

The one exception is instructive. At the finest rise on the rung, offset four keeps the eight rather than the five, which is the row that changes hands along the rung. It is a silent row by the divisibility test and it is also the single row in the whole sweep whose answer depends on the rise. Whatever is happening at the offsets that belong to no chain is evidently more sensitive to the geometry than what happens at the offsets that do — which is a hint about where to look, and is as far as a hint should be pushed.

The offset rule covers them by arithmetic: four is no further back than five, so the five survives; six and seven are beyond it, so the eight does. That is a correct description of what happens and it says nothing about why an organ that is nobody’s neighbour should have any effect at all.

And it should have an effect — that part is not mysterious. The rule minimises a sum over every organ already placed, not over a chain, so a removal anywhere inside the front changes the sum the next organ is placed against whether or not the removed organ was anybody’s nearest neighbour. What is mysterious is why the outcome should sort itself into two families when the disturbance does not respect chains at all. The restriction to two families is the strongest result this thread has, and the twenty-one are where it is least explained.

The extension that is available, and why it is not taken #

There is an obvious way to make the reading cover everything, and it is worth setting out precisely so that its cost is visible.

Chains are not a property of the tip. The organ four places back lies on the five-chain through the organ one place above the tip, and on the eight-chain through some organ further down; every organ lies on both chains through somebody. So one could say: the family that survives is the one whose chain through the nearest affected organ lost a member — and with the right choice of reference organ, every row becomes answerable.

That is exactly the shape of a free parameter. “The nearest affected organ” is a choice, and there are several defensible ones — the tip, the newest organ whose placement moved, the organ closest to the hole in the arrangement rather than in the sequence. Each gives a different assignment on the twenty-one rows, and with three choices and twenty-one rows it would be surprising if none of them scored well.

This collection has been caught by exactly that shape before. Four different definitions of how deep the rule looks disagree about the size of the neighbourhood by a factor of ten, and the choice between them was a free parameter nobody had varied. A reading that needs a reference organ chosen after the rows are known is a reading whose score is about the choosing.

What would earn the extension #

A reference organ chosen for a reason rather than for a score, and stated before the rows are looked at.

There is a candidate with a reason behind it. The rule places each organ at the minimum of a sum over the organs already there, so the organ whose placement the removal actually disturbed most is a measurable thing, not a choice: it is the one whose azimuth moves furthest between the cut run and its control.

Defining the reference organ that way is falsifiable in the way the tip version is: compute the most-disturbed organ, ask which chains it lies on, and score the prediction on all thirty rows at once. If it holds on the twenty-one as well as the tip version holds on the nine, the reading is the account. If it does not, it is dead, and it dies on rows chosen by a rule rather than by hand.

That measurement does not exist yet. It is one column, like the last one, and this essay’s job is to say what the column has to contain rather than to guess what it will say.

There is a reason to expect it to be interesting rather than merely confirmatory. The organ whose placement moves most is not always adjacent to the hole: the response to a removal is not a smooth falloff along the sequence but a pattern with structure in it, because the sum being minimised is over positions in the arrangement rather than over indices. So the most-disturbed organ could easily lie several places from the cut, on chains the cut itself is not on, and the prediction it generates need not agree with the offset rule at all. A reading that can disagree with the incumbent is worth computing; one that cannot is bookkeeping.

Three things the twenty-one already rule out #

They are not a ragged remainder. They are the offsets strictly between the two counted numbers, plus those below the smaller one, and they behave consistently: below the smaller number they keep the smaller family, between the two they keep the larger. Whatever is happening is systematic.

They are not explained by step length. The shortest-hop reading is a coin flip along a rung, and restricting it to these twenty-one does not improve it.

And they are not an artefact of one lattice. The twenty-one are spread across five lattices on two branches, and the pattern is the same on both. Nor are they concentrated at one end of the front: they run from offsets below the smaller counted number to offsets between the two, and both groups behave consistently with the offset rule while being invisible to the lost-member one. If the silent rows were all crowded against the deepest edge of the front — where a removal is barely felt and the wreck is marginal — the gap could be dismissed as a boundary effect. They are not, and it cannot.

Why a gap is worth publishing #

A collection that only published the nine would be publishing a rule with a hidden domain, and hidden domains are how a fit becomes folklore. The nine are a strong result — a stated prediction, refuted backwards, replaced by its opposite at every row — and they cover three tenths of the census. Both halves of that sentence are load-bearing.

The failure mode is specific and this thread has already produced one instance of it. “The rule keeps its shortest hop” was stated as a rule and scored at twelve of twenty-nine, which sounds like a refutation and was really a statement about a census whose sampling correlated the ordering with the pair. A reading quoted without its domain is a reading whose next reader cannot tell which of those two situations they are in. Writing the twenty-one down is what makes the nine quotable.

There is a sharper reason than tidiness, and it comes from setting this essay’s count beside the one the offset rule carries.

The offset rule is right at twenty-five of thirty. The lost-member reading is right at nine of nine — and on those nine the two return the same answer, because an organ lying on the tip’s p-chain is also, in the offset rule’s arithmetic, a cut landing at or beyond p. So the five rows that defeat the offset rule cannot be among the nine. All five of them are inside these twenty-one.

That changes what the gap is. It is not a region where a good rule has yet to be extended. It is the region where the best available rule does all of its failing, and where the mechanism that might explain those failures is silent by construction. The nine rows the two accounts share carry no information whatever about their disagreement.

It also prices the cheap strengthening described above. Sweeping finer rungs adds answerable rows, and answerable rows are rows where the two accounts already agree, so thirty or forty of them would raise the lost-member reading’s count without touching one row that separates it from the arithmetic it is meant to explain. As a check against coincidence the strengthening is real; as a test between the two readings it is worth nothing.

That test has to be run on these twenty-one, which is the whole case for treating them as the subject rather than the leftovers. A census built to settle the thread would sample offsets that are nobody’s neighbour on purpose, at several rises, and ask whether the answers there follow anything at all — because if they follow nothing, the offset rule’s five failures are not failures of a rule but the visible edge of a domain nobody has drawn.

The gap widens with every rung climbed #

The obvious way to shrink twenty-one is to sweep finer rungs, and the arithmetic above says it does the opposite.

Each lattice contributes two answerable offsets and q − 2 silent ones, where q is the front. At 3/5 that is two answerable and three silent. At 5/8 it is two and six. At 8/13, two and eleven. At 13/21, two and nineteen. The answerable share falls from two fifths to under a tenth as the ladder is climbed, and the count of silent rows per lattice grows without limit while the count of answerable ones never changes.

So a census assembled at finer rungs would be a census in which the lost-member reading is silent almost everywhere. The nine would grow only in proportion to how many lattices were swept, never to how deep their fronts are, and each new lattice would bring five, eleven or nineteen rows the reading cannot address against the two it can.

That inverts the plan. The way to more answerable rows is more lattices at whatever rung is cheapest — and the cheapest lattices are the twelve inside a single rung, each carrying its own two, which is twenty-four answerable rows off one sweep of the coarsest rung that wrecks at all. Climbing the ladder buys the reading nothing and buys its gap a great deal.

The silent rows are therefore a permanent feature of asking the question by divisibility rather than a backlog to be worked off, and they get worse rather than better as the collection moves to the arrangements it has not yet visited. Any account that finishes this thread has to speak to them directly. Sweeping along a rung adds rows of both kinds at once, which is the cheapest way to find out whether the two behave differently under the one variable the census never moved.

The chains govern the silent rows too, through the boundary #

There is one thing the twenty-one do say about chains, and it is easy to miss because it is a statement about where a behaviour changes rather than about any row.

On the silent offsets the answers sort themselves cleanly: below the smaller counted number the smaller family survives, between the two numbers the larger one does. Nothing about those organs is a chain member. But the place where the behaviour changes is the smaller counted number itself — not four and a half, not somewhere between six and seven, and not at the middle of the front. It is at p, the same number the answerable rows are built out of.

So the counted numbers are doing work on rows where no removed organ lies on a chain. What they are not doing is the work the lost-member reading describes, which is about membership. They are setting a threshold instead, and a threshold at p is a fact about the arrangement that a reading stated in terms of membership has no way to express.

That is a real constraint on what the missing account can look like. It cannot be a reading that only speaks about chain members and shrugs at everything else, because the boundary between the twenty-one rows’ two behaviours sits exactly on a chain number and would have to be a coincidence otherwise. And it cannot be a reading with no chains in it at all, because p is a chain number and nothing else about the front picks it out — the front’s own depth is q, the repaired edges are three organs, and neither of those lands anywhere near p at more than one rung.

What fits is a reading about which chain the next organ is being placed against, rather than which chain lost a member. An organ removed no further back than p is inside the stretch of history the tip’s p-chain occupies; one removed beyond it is not. That is a statement about the removal’s position relative to a chain rather than about its membership of one, it covers the silent rows and the answerable ones in the same sentence, and it is testable in the obvious way — by finding an arrangement where the two formulations disagree.

Where this leaves it #

The thread now has a mechanism-shaped statement that is right wherever it applies, an arithmetic rule that covers nearly everything and explains nothing, and a clean boundary between them. That is a better position than a single rule at twenty-five of thirty, because the boundary is where the next measurement goes.

Nothing here is a placeholder for a result that is coming. It is a statement of what twenty-one rows do not currently support, written down so that the next person to score a reading over this census — including a later version of this collection — has to score it over all thirty.

The general form is worth keeping. A reading that applies to a subset defined by the data rather than by the question is a reading with a domain, and the domain has to be stated in the same breath as the score. Here the subset is defined by divisibility, which is at least a property of the arrangement and not of the outcome — so the nine are a fair test rather than a selection. Had the subset been “the rows where the reading works”, there would be no result at all, and the distance between those two situations is exactly one sentence of bookkeeping that is easy not to write.